Lesson 1: Functions

Sections 1.1–1.2
Functions

V63.0121, Calculus I

September 10, 2009

Announcements
Syllabus is on the common Blackboard
Ofﬁce Hours TBA

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Outline
What is a function?
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
. . . . . .

Deﬁnition
A function f is a relation which assigns to to every element x in a
set D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { f(x) | x ∈ D } is called the range of f.

. . . . . .

The Modeling Process

. .
Real-world
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

. . . . . .

Plato’s Cave

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The Modeling Process

. .
Real-world
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

S
. hadows F
. orms

. . . . . .

Functions expressed by formulas

Any expression in a single variable x deﬁnes a function. In this
case, the domain is understood to be the largest set of x which
after substitution, give a real number.

. . . . . .

Example
x+1
Let f(x) = . Find the domain and range of f.
x−1

. . . . . .

Example
x+1
Let f(x) = . Find the domain and range of f.
x−1
Solution
The denominator is zero when x = 1, so the domain is all real
numbers excepting one. As for the range, we can solve

x+1 y+1
y= =⇒ x =
x−1 y−1

So as long as y ̸= 1, there is an x associated to y.

. . . . . .

No-no’s for expressions

Cannot have zero in the denominator of an expression
Cannot have a negative number under an even root (e.g.,
square root)
Cannot have the logarithm of a negative number

. . . . . .

Piecewise-deﬁned functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.

. . . . . .

Piecewise-deﬁned functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.

Solution
The domain is [0, 2]. The range is [0, 2). The graph is piecewise.

. .
2 .

. .
1 . .

. . .
0
. 1
. 2
.

. . . . . .

Functions described numerically

We can just describe a function by a table of values, or a diagram.

. . . . . .

Example

Is this a function? If so, what is the range?

x f(x)
1 4
2 5
3 6

. . . . . .

Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6

. . . . . .

Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6

Yes, the range is {4, 5, 6}.

. . . . . .

Example


x f(x)
1 4
2 4
3 6

. . . . . .

Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6

. . . . . .

Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6

Yes, the range is {4, 6}.

. . . . . .

Example

How about this one?

x f(x)
1 4
1 5
3 6

. . . . . .

Example

How about this one?

. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6

. . . . . .

Example

How about this one?

. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6

No, that one’s not “deterministic.”

. . . . . .

In science, functions are often deﬁned by data. Or, we observe
data and assume that it’s close to some nice continuous function.

. . . . . .

Example
Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8

. . . . . .

Sometimes all we have is the “picture” of a function, by which
we mean, its graph.

.

.

. . . . . .

Sometimes all we have is the “picture” of a function, by which
we mean, its graph.

.

.

The one on the right is a relation but not a function.

. . . . . .

Functions described verbally

Oftentimes our functions come out of nature and have verbal
descriptions:
The temperature T(t) in this room at time t.
The elevation h(θ) of the point on the equation at longitude
θ.
The utility u(x) I derive by consuming x burritos.

. . . . . .

Monotonicity

Example
Let P(x) be the probability that my income was at least $x last
year. What might a graph of P(x) look like?

. . . . . .

Monotonicity

Example
Let P(x) be the probability that my income was at least $x last
year. What might a graph of P(x) look like?

. .
1

.

. . . . . .

Monotonicity

Deﬁnition
A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2
for any two points x1 and x2 in the domain of f.
A function f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2
for any two points x1 and x2 in the domain of f.

. . . . . .

Examples

Example
Going back to the burrito function, would you call it increasing?

. . . . . .

Examples

Example
Going back to the burrito function, would you call it increasing?

Example
Obviously, the temperature in Boise is neither increasing nor
decreasing.

. . . . . .

Classes of Functions

linear functions, deﬁned by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions

. . . . . .

Linear functions

Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.

. . . . . .

Linear functions

form
f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.

. . . . . .

Linear functions

form
f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.

Answer
If x is in miles and f(x) in dollars,

f(x) = 2.5 + 2x

. . . . . .


Quadratic functions take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0,
downward if a < 0.

. . . . . .


Quadratic functions take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0,
downward if a < 0.
Cubic functions take the form

f(x) = ax3 + bx2 + cx + d

. . . . . .

Other power functions

Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.

. . . . . .

Rational functions

Deﬁnition
A rational function is a quotient of polynomials.

Example
x 3 (x + 3 )
The function f(x) = is rational.
(x + 2)(x − 1)

. . . . . .

Trigonometric Functions

Sine and cosine
Tangent and cotangent
Secant and cosecant

. . . . . .

Exponential and Logarithmic functions

exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example
f(x) = log2 (x))

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Lesson 1: Functions

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Lesson 1: Functions